NAVAL POSTGRADUATE SCHOOL · PORTSDIVI •TESCHO( 93941 NPS55-79-020 NAVALPOSTGRADUATESCHOOL Monterey,California STATISTICALMETHODSOFPROBABLEUSE FORUNDERSTANDINGREMOTE by SENSINGDATA

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STATISTICAL METHODS OF PROBABLE USE

FOR UNDERSTANDING REMOTE

by

SENSING DATA

Donald P. Gave r

October 19 79

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Statistical Methods of Probable Use for Under-standing Remote Sensing Data

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StatisticsRobustnessRemote SensingRegression

CensoringSmoothingExtreme Values

20. ABSTRACT (Continue on reverae aide If necessary and Identity by block number)

This report outlines several new statistical approaches to data problemslikely to be encountered when remote sensing methods are used. Themethods described are robust regression, smoothing, and modeling andestimation of ice pressure ridge characteristics.

dd ,;FORMAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETE

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SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

STATISTICAL METHODS OF PROBABLE USE

FOR UNDERSTANDING REMOTE SENSING DATA

Donald P. Gaver

Naval Postgraduate SchoolMonterey, CA 9 3940

1. INTRODUCTION

Statistical methodology has long been a familiar tool for

use in understanding our natural environment. Classical examples

of applications of statistics are seen in weather forecasting,

in evaluation of attempts at weather modification by cloud seed-

ing, and in descriptions of the fluctuations in the sea surface.

Now the accessibility of new and extensive data from a variety

of remote sensing sources, such as earth orbiting and geostationary

satellites, again calls for the development and application of

appropriate statistical methodology. Classical methods of statis-

tics and of probability modeling frequently must be adapted to

the new needs. The process of adaptation will proceed most

efficiently if statisticians work cooperatively with the scientists

actually obtaining data and studying the associated natural

phenomena. Conferences such as PRIMARS I are of great value in

promoting the necessary interchange of information and the stimulus

to approach novel and difficult problems in a realistic manner.

This paper describes new approaches to the analysis of

data, in particular to quite "noisy" data of the sort that is

likely to be encountered when observing the natural environment.

The descriptions given will necessarily be brief, but an attempt

will be made to show how the methods and viewpoints presented may

be applied to problems arising in remote sensing.

2. ROBUST METHODOLOGY: REQUIREMENTS AND POSSIBILITIES

Many scientists who have closely examined real data

have encountered occasional, or even frequent, anomalous behavior.

Apparent anomalies in data may be with respect to either

(a) preconceptions as to "proper" data behavior, these perhaps

being buttressed by (physical) theory, o£

(b) the nature of the general pattern of the data, especially

those data points in the immediate neighborhood, e.g.,

in time or space.

2.1. Plots

In simple circumstances graphical plots will quickly reveal

those points that are blatant anomalies. For instance suppose that

one wishes to investigate data concerning the relationship between

wind velocity and whitecap cover in the ocean. Theory may suggest

a specific relationship, e.g. that white-cap cover, C, be nearly

a cubic function of wind velocity v, so that it will be tempting

to plot C vs v and note an appearance as shown on Fig. 1; there

solid black dots represent (simulated) raw data. Since the eye

2finds it difficult to distinguish curves of the form C = av ,

3 7/2 •

C = av , C = av ', etc., froir. one another, and yet is sensitive to

3departures from linearity, a graph of C vs v suggests itself, bu1

is not included here. A plot on log-log paper may be still better.

As presented, the data conforms in general to the theorized relation-

ship or scaling, with the obvious exception of the circled point to

the right. Such an anomalous point, or points, represents a challenge

both to statistical technology and to the ultimate user of the data.

Statistical technology assumes the responsibility for revealing

the presence of such points, and, if possible, for providing a

meaningful and useful summary of the remaining points. It falls

to the consumer or ultimate user of the data, preferably with

the help of a subject-matter specialist (physicist or oceanographer)

to interpret the apparently anomalous maverick—or exotic, or

outlying—data point: is it

(i) an evidence of the failure of the relation C = av , say

for large velocities,

or is it

(ii) an outright error in data recording, and to be disregarded?

being just two possible options.

Note that simple graphs are invaluable for pointing out

extreme outliers in simple, one explanatory variable, situations.

If more variables are required, informative plots are more diffi-

cult without the use of more statistical technology. We next

show that classical, least-squares, technology may be quite mis-

leading, but that replacements are available. See Mosteller and

Tukey (19 77) , abbreviated MT hereafter.

2.2. Fits and Residual Plots

Suppose that one wishes to summarize data such as that

in Fig. 1 by fitting the relationship C = av , i.e., determining

the parameter a from the data. The classical and automatic way

of doing so is to apply least squares; computer programs are uni-

versally available, even for handheld calculators (the TI 59, or

HP 67) . What are we likely to find? A least-squares line (treat-

3ing w = v as the independent variable presents C = aw; one

can also plot and fit C '* = (a) ' v, and there may be reasons

for this choice) is quite apt to fatally misrepresent the situation,

responding much too sensitively to the single (here encircled)

outlying value, and straying systematically away from the main

body of the data; see the points represented by o in Fig. 1.

An alternative method for fitting, described in MT, is

less susceptible to outlier influence— is far more robust to

departures from basic assumptions— than is the ordinary least

squares (OLS) method. This new method*, termed biweight fitting ,

is carried out by a procedure that uses the OLS computation itera-

tively. In the course of the computations weights are auto-

matically developed that reduce the influence of the encircled

value of Fig. 1, permitting the fit to more closely approximate

the main body of the data. We now describe and illustrate the

biweight fitting procedure as it is adapted to the problem of

determining the parameter a in the relation y. vs_ ax..

Biweight Fitting Calculation

(1) Compute the kth (k = 1,2,3,...) iterative estimate of a,

denoted by a by solving

n) (y. - a x. ) x.w.

(k-1)= ,

to obtain

n(k-1)

a

y y . x. w.(k> _ i=i x x *

(k-1)(2) the weights, w. , are of this form:

w (k-1)1 -

(k-1) \2y. - a x.

cSTFTTif (') < 1

= if (•) > 1

where (•) refers to the term [(y.-a (k 1)x. ) /S

(k~ 1)]

;

(k-1)S is a scale factor (robust replacement for the

standard deviation) that may be computed in the following manner

s t(3) The k-1 iterated value of the scale factor is

c (k-l) ,. n (k-1) ,,S = median! |y. - a x.|>,

c being a constant of value 6, or 9;

(4) the first value, a , of the iterative sequence can be

obtained by equalizing all weights (w. = 1) , which is

equivalent to OLS ; alternatively, one can utilize a "robust

start," suggestions for which can be found in MT .

The iteration is carried on until the difference between successive

values is small; usually 4 to 8 iterations is sufficient. The

resulting a-estimate can be denoted by a.

Following the fitting it is informative to plot the

residual values:

r. = y- - ax. =y. - y. , i=l,2,...,n,

y. being shorthand for the predicted y value. In case there

is a single outlier, as in Fig. 1, the fitted line will tend to

hug the major point cloud, and a histogram of the residuals will

dramatically reveal the presence of the outlier, suggesting

further investigation. A plot of r. vs y. is also useful. See

MT for further suggestions

.

2.3. Numerical Illustration

The following are a set of (simulated) whitecap percentages

and corresponding wind velocities. Alongside are values for

white cap coverage estimated by OLS and by the biweight procedure

.

Veloci ty Cover("Actual"

2 0.011

7 0.63

10 1.30

15 3.89

18 2.89

21 8.16

24 25.7

Cover(OLS Estimate)

0.011

0.49

1.43

4.82

8.33

13.2

19.7

Cover(Robust Estimate)

0.0067

0.29

0.84

2.83

4.88

7.76

11.6

It is clear from the above table, and perhaps clearer from Fig. 1,

that the OLS solution, in its attempt to fit the point C(24) = 25,

systematically and considerably over-estimates the points at v = 15

and above. The biweight estimator performs much better, allowing

a closer fit to all data other than C(24) . A residual plot brings

attention to bear on that point.

Since the values of "Actual Cover" were actually constructed

3by forming 0.00 8v and adding Gaussian random noise with value pro-

3portional to C(v), and since the sequence of values of 0.008v

were 0.0064, 0.27, 0.80, 2.7, 4.67, 7.41, 11.06, we cannot fault

the manner in which the biweight procedure functioned in this

example and are encouraged to use it more widely.

2.4. Possible Application to Remote Sensing Data

In a paper in this conference proceedings by Depriest

(1979), and in Fleming (1979), a problem arising from partial

cloud cover contamination of remote sensing data is described

and addressed. This problem has the following origin. A series

of measurements are made on a physical quantity (sea surface

temperatures) but are contaminated. That is, in the case of

sea surface temperatures, if no clouds are present the measure-

ments are approximately normally distributed around y (the true

temperature) . However, if clouds are present a fraction of the

measurements are made artificially smaller, cloud temperatures

being lower than those at earth surface. The problem is to esti-

mate y. Techniques for doing so are described by Depriest (1979)

and by Fleming (19 79) . We describe a possible alternative approach

that uses robust regression. Operational characteristics of

the two procedures have not yet been compared.

(1) Arrange the measurements in order: y < y < y < ... <J- £• J

yn-l < yn*The lar9est observations may well appear

similar to the largest order statistics of a normal distri-

bution with (unknown) mean y and standard deviation a

(sometimes assumed known, although caution is in order)

,

while the smaller ones are likely to depart systematically.

(2) Carry out a preliminary plot of

yk XS*" 1

(fÎ ) , k = n, n-1, n-2,

where * (p) is the inverse function of the unit normal

recall that if

* / \ f / 1 2

.

dz* (y) =

J expC-j z )——- ,

-°° /2tt

is the unit normal distribution/ then the solution of the

equation $ (y) = p gives

y(p) = $~ (p)

;

$, and hence $ , are widely tabulated. Alternatively,

use Arithmetic Probability Paper. If y, is an ordered

observation from a normal population, then the plot should

appear straight, while a systematic departure from linearity

indicates a departure from normality. Suppose departures

begin to occur at k = D; sometimes D may be greater than

n/2 . One may first eye-fit a straight line to the points

k = n, n-1, ... , D. Then y ,~ = u (estimated temperature).

i.e. the value of the fitted line at n/2 should give a

reasonable value for y.

(3) Going further in a formal direction, one may wish to fit a

line to the data points. Here a biweight fit should behave

well, tending to be oblivious to spurious (cloud contamina-

tion) points. One can proceed to fit the relation

ykvs u + ax

k

with

-1 kXk

= ° ^n+T* ' k = n, n-1, n-2, ... ;

a start using the eye-fit to points k = n, n-1, ... , D

may be worthwhile. Finally, quote the estimate

y = med yk= | (y + y (n/2)+1 >, n even;

= y (n+l)/2'n odd '

where

yk= V + ax

k.

The above procedure seems worth further investigation and refine-

ment. One important step may be to adjust for the effect of

correlation between order statistics when carrying out the

regression.

10

3. SMOOTHING DATA

If one plots certain environmental data, e.g. monthly

total rainfall, or perhaps daily maximum temperature, at a

particular location, systematic regularities seem to appear,

but may be masked by noise. Often there is a seasonal pattern,

i.e. one that is roughly cyclic in nature. Attempts to fit such

a pattern with polynomials is doomed to failure, and selection

of a set of sines and cosines that does well (Fourier series)

may lead to many terms. Some method of smoothing the original

series that lays bare the regularities is to be desired. After

such is made available, one can study the residuals around it.

Spectral analysis or some such formal procedure may then be of

use.

Classical smoothing procedures involve some form of moving

average, and are susceptible to the python-swallowing-the pig

difficulty: imagine using the linear smoothing operation

yt-i+ yt

+ yt+iSY<t 3

on the y series

t 1234 5 6 789 10

y 11 9 7 8 8 29 10 8 6 9

Syt <Q 9 8 7.7 15 15^J7 15^7 8 7.7 @

Ry. (fi) 9 8 8 8 10 10 88©

11

clearly the entry of 29 at t = 5 into the smoothed value

series Sy gives a serious distortion, travelling as it does

in partially digested form through the next two terms of the

smoothed series. If further smoothing is attempted the bulge

is reduced slightly, but spreads out in time.

On the other hand, the non-linear operation of taking

running medians, as suggested by Tukey (1977)/ performs effec-

tively (in both cases circled and values are copied from the

original series; more sophisticated procedures can be invented

as well). The last row in the table, labelled Ry , gives the

result of this robust smoothing; note that it behaves in an

intuitively appealing manner, essentially ignoring the outlying

value 29. Further steps can be taken to improve the "smooths,"

but we refer to Tukey (19 77) Chapters 7 and 16 for details.

Two further points may be made. The first is that the

analysis of a sequence of data points, and their projection or

forecasting in space or time, should not end with providing a

smoothed or averaged version. Examination of the remaining vari-

ation, e.g. the sequence y - Ry , called the "rough" by Tukey

may well be rewarding; presence and suggestiveness of various

outliers is much more evident in the rough (residuals) sequence

than in the original sequence of data points. Secondly, the

procedure described for smoothing simple sequences of data points

must be adapted to planar (two-dimensional) data; some work has

been done, but much remains.

12

4. STOCHASTIC MODELING OF ICE PRESSURE RIDGES

The dynamics of ice formation in the Earth's cold regions

results in the development of irregular ice pile-ups, or pressure

ridges. These ridges occur in an apparently random fashion in

space; in fact the following regularities are observed by remote

sensing methods (courtesy of Dr. W. Weeks, in a seminar at the

Naval Postgraduate School, Monterey, California, Winter, 1979;

see also Weeks, et al. (1979)):

1) Along a sampling line (e.g. airplane flight path, or straight

submarine track) ice ridges seem to appear in accordance

with a stationary Poisson process, so if R(x) is the number

of such ridges encountered over a distance x, then approxi-

mately

P{R(x) = n} = e"Xx (X* }

, , n = 0,1,2,...n •

where A > is the density of ice ridges.

2) The probability distribution of ridge "sail heights" (or

"keel depths") may be approximated by the forms F(y) = 1-e2

or 1-e y; the best-fitting distribution may well depend

upon the method of observation (averaging properties).

For further details see work referenced in Weeks et a_l. (1979) .

Now it may be of interest to compute the distribution

of the maximum sail height, or keel depth, that one is to encounter

over a course of length x. This is very simple, given the

particular distributions of -sail number and size and furthermore

13

assuming independent between ridge heights. Let H(x) be the

maximum sail height; then

n-Ax (Ax)

r_ .

x,n

P{H(x) £ y> =I e"

AX A^_[F (y)l

n=0 n '

since all of the Poisson-distributed heights must be less than y

in order for the maximum to be below y.

Sum out to obtain

P(H(x) <_ y) = exp{-Ax[l-F(y) ]}

Depending upon which distribution is picked for ridge heights,

we get

a) P(H(x) £ y} = exp(-Axe~yy )

2

b) P(H(x) <_ y) = exp(-Axevy

)

These closely resemble classical extreme value distributions.

Note that if logs are taken simplicity occurs:

a') £n p{H(x) £ y) = -Axe' 'VY.

£n(-£n P(H(x) <_ y}) = Zn(Xx) - yy

2

b') £n P{H(x) £ y} = -Axe"Vy

;

£n(-£n P{H(x) y}) = £n(Ax) - vy2

14

If either of these formulas are to be used for practical purposes,

values of the parameters must be obtained. In order to estimate

parameters X, u, \> in the above models from data one naturally

thinks of the method of maximum likelihood. Suppose that we

have observed R(x) = n ridges of heights y 1# y 2, •• ' ¥n

*

Then the maximum likelihood estimates are

x * 1 * _ _1_t u ~ T ' v

vX - n ' M - - ' v

y 2

where as usual we have put

k 1 f k

i=l

Hence our estimates are of the form

a") est £n(-£n P{H(x) <_ y} ) = in A + in x - yy

b") est ln(-ln P{H(x) <_ y} ) = In A + £n x - vy

If rather large samples are available and if distributional assump-

tions are well satisfied one may feel comfortable with conventional

standard errors based on Fisher information and normality; see

Cramer (1946). On the other hand, it is of interest to apply

the jackknife technique (see R. G. Miller (1974) for a review) to

obtain estimates of the variance of estimate due particularly to

the ridge heights. To carry out the calculation, (i) compute

v ,, = n/y ; then (ii) compute

n-1v(-j) 2 2 2

" 2 T~y, + y-> '+•••+ y. , + o + v. .

+•••+ y

15

for j = 1,2,..., n; then (iii) compute the pseudovalues

v. = nv ., - (n-1) v. . ), and (iv) average to obtain a jackknifed3 all v~D

por n

int estimate vtv = (1/n) ) . . v., and its varianceJK L j=l j

Then we can estimate the standard error of the probability predictior

e.g. b") by computing

S.E. = (Var[est £n(-£n P(H(x) < y})])1/2

A similar calculation is easily performed for model a) ; details

are omitted. From the above results, approximate confidence inter-

vals may be constructed for the probability of encountering a

(maximum) ridge sail height less than y in magnitude.

Fairly recent theoretical results of Efron and Hinkley

(1978) suggest that if a traditional maximum likelihood approach

is taken, one is better off using observed Fisher information

rather than expected Fisher information in order to establish an

approximate standard error in either case a) or b) . However,

work of Reeds (1978) suggests that use of the jackknife in

conjunction with maximum likelihood yields results that tend

16

to be rather independent of the basic model chosen. Both of

these suggestions must be validated by further work, a good

deal of which will necessarily involve Monte Carlo simulation.

Such work should be of great importance and interest to those

who must assess the probabilities of extreme, rare, events, and

who furthermore wish to provide some reasonably valid estimates

of the error of their estimates.

Acknowledgment . The writer is much indebted to LCDR C. F.

Taylor, Jr., for his assistance in example robust regression

computations. He is also indebted to the Office of Naval

Research for support of this research.

17

REFERENCES

Cramer, H. (1946), Mathematical Methods of Statistics , PrincetonUniv. Press, Princeton, N.J.

Depriest, D. (1979) , "Consideration using a truncated normaldistribution for remote sensing data." Paper presentedat PRIMARS-1 Conference.

Efron, B., and Hinkley, D - (1978). "Assessing the accuracy ofthe maximum likelihood estimator: observed versus expectedFisher information," Biometrika 65 , No. 3, pp. 457-488.

Fleming, H. E. (1979). Application of the truncated normal dis-tribution technique to the derivation of sea surface tem-peratures," to appear in Remote Sensing of Atmospheres andOceans (1980), ed. by A. Deepak; Academic Press.

Miller, R. G. (1974). "The Jackknife— a review." Biometrika 61,No. 1, pp 1-16.

Mosteller, F. , and Tukey, J. W. (1977) , Data Analysis andRegression . Addison-Wesley Publishing Co., Reading, Mass.

Reeds, J. A. (1978). "Jackknifing maximum likelihood estimates."Annals of Statistics 6, No. 4, pp. 727-739.

Tukey, J. W. (1977). Exploratory Data Analysis , Addison-WesleyPublishing Co., Reading, Mass.

Weeks, N. F., Tucker, W. B., Frank, M. , and Fungcharoen, S . (1979)."Characterization of the surface roughness and floe geometryof the sea ice over the continental shelves of the Beaufortand Chukchi seas." In "Sea Ice Processes and Models, Proc.AIDJEX/ICSI Sympos." (R.S. Pritchard, ed.), University ofWashington Press (in press)

.

18

25

WHITECAP COVERAGE vs. WIND SPEED( SIMULATED DATA )

• DATAO ORDINARY LEAST-SQUARESD ROBUST (Bl WEIGHT) FIT

FIT (a = 0.00114 )

(a = 0.00837 )

®

20 -oIDe><LU>o<J

a.<oUJ

I

15

a

10

o

a

a

I—0- ti-

2a

o 15

Fig. 1

20 25 30' velocity \

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